Customer Reviews

Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics)

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Liu's book has two distinct parts to it. The first 7 chapters combine to give a wonderful exposition of the language of schemes; the other chapters are of a specialised nature and concentrate on arithmetic curves. I will talk about the former. (So when I say "this book", I am only referring to the first 7 chapters)

The book starts off with a chapter on some topics in basic commutative algebra - localisation, flatness and completion. Once this is done, the stage is set to introduce schemes in the next chapter and prove their basic properties. Chapter 3 talks about morphisms of schemes and base change. Chapter 4 continues with a discussion of morphisms and also presents some results about some special types of schemes (normal, regular). It culminates with a proof of Zariski's main theorem. The next chapter takes up sheaf cohomology and is followed up with a chapter on differential calculus on schemes (Kahler differentials, duality theory). Lastly, chapter 7 takes up divisors, proves the Riemann Roch theorem and culminates with some applications to curves.

At a first glance, this would basically look like Hartshorne - the most popular book for an introduction to schemes. However, there are few differences which I will point out. Firstly, Hartshorne emphasizes geometric applications and, as such, uses algebraically closed fields freely. Liu, on the other hand, does not hesistate to give arithmetic applications whenever possible and, therefore, tries to relax the hypotheses on the base field whenever possible. Secondly, Liu is much more readable than Hartshorne which, in its supreme elegance, is a tad dense for a first reading. Unlike Hartshorne, a majority of important results are not presented in the exercises (though many are!). Moreover, unlike Harshorne, this book develops all the necessary commutative algebra along the way (chapter 1,2 of Atiyah-Macdonald should be good enough to read this book). Coming back to the geometry, Hartshorne's chapter 4,5 form an excellent resource for classical geometric applications for theory of schemes. Moreover, chapter 1 presents a very readable and scheme-free account of classical algebraic geometry (pre-Grothendieck) in the language of varieties. Liu's book, however, does not emphasize classical or geometric applications and is not the best place to start if one wishes to learn about varieties.

In the current literature on algebraic geometry, there is a noticeable void. Namely, on one hand, we have Grothendieck's "Elements" (EGA) which present all results about schemes and sheaf cohomology in utmost generality, prove everything with excruciating detail, and are almost unreadable as texts (they're a great references). On the other hand, we have Hartshorne which is basically a beautiful summary of EGA along with geometric applications, but is quite hard to read for an introduction. The book under review is not as concise as Hartshorne's book, presents arithmetic applications and is more readable in a reasonable amount of time than EGA.

In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind. For those inclined towards geometry, an account of schemes from this book coupled with applications from another book (like Hartshorne) would be a good combination.

This book together with Matsumura on Commutative Algebra and Hartschone on Algebraic Geometry is an excellent book to learn the subject. I am really enjoying it.